Academic literature on the topic 'Eshelby Equivalent Inclusion Method (EIM)'

A micromechanical approach is developed to investigate the behavior of composite materials, which undergo interfacial delamination. The main objective of this approach is to build a bridge between the intricate theories and the engineering applications. On the basis of the spring-layer model, which is useful to treat the interfacial debonding and sliding, the present paper proposes a convenient method to assess the effects of delamination on the overall properties of composites. By applying the Equivalent Inclusion Method (EIM), two fundamental tensors are derived in the present model, the modified Eshelby tensor, and the compliance tensor (or stiffness tensor) of the weakened inclusions. Both of them are the fundamental tensors for constructing the overall constitutive law of composite materials. By simply substituting these tensors into an existing constitutive model, for instance, the Mori-Tanaka model, one can easily evaluate the effects of interfacial delamination on the overall properties of composite materials. Therefore, the present method offers a pretty convenient tool. Some numerical results are carried out in order to demonstrate the performance of this model.

Consider a double-inhomogeneity system whose microstructural configuration is composed of an ellipsoidal inhomogeneity of arbitrary elastic constants, size, and orientation encapsulated in another ellipsoidal inhomogeneity, which in turn is surrounded by an infinite medium. Each of these three constituents in general possesses elastic constants different from one another. The double-inhomogeneity system under consideration is subjected to far-field strain (stress). Using the equivalent inclusion method (EIM), the double inhomogeneity is replaced by an equivalent double-inclusion (EDI) problem with proper polynomial eigenstrains. The double inclusion is subsequently broken down to single-inclusion problems by means of superposition. The present theory is the first to obtain the actual distribution rather than the averages of the field quantities over the double inhomogeneity using Eshelby’s EIM. The present method is precise and is valid for thin as well as thick layers of coatings, and accommodates eccentric heterogeneity of arbitrary size and orientation. To establish the accuracy and robustness of the present method and for the sake of comparison, results on some of the previously reported problems, which are special cases encompassed by the present theory, will be re-examined. The formulations are easily extended to treat multi-inhomogeneity cases, where an inhomogeneity is surrounded by many layers of coatings. Employing an averaging scheme to the present theory, the average consistency conditions reported by Hori and Nemat-Nasser for the evaluation of average strains and stresses are recovered.

This paper develops a three-dimensional (3D) model for a heterogeneous half-space with inclusions distributed periodically beneath its surface subject to elastohydrodynamic lubrication (EHL) line-contact applied by a cylindrical loading body. The model takes into account the interactions between the loading body, the fluid lubricant and the heterogeneous half-space. In the absence of subsurface inclusions, the surface contact pressure distribution, the half-space surface deformation and the lubricant film thickness profile are obtained through solving a unified Reynolds equation system. The inclusions are homogenized according to Eshelby’s equivalent inclusion method (EIM) with unknown eigenstrains to be determined. The disturbed half-space surface deformations induced by the subsurface inclusions or eigenstrains are iteratively introduced into the lubricant film thickness until the surface deformation finally converges. Both time-independent smooth surface contact and time-dependent rough surface contact are considered for the lubricated contact problem.

The self-similarly dynamically (subsonically) expanding anisotropic ellipsoidal Eshelby inclusion is shown to exhibit the constant stress “Eshelby property” in the interior domain of the expanding inclusion on the basis of dimensional analysis, analytic properties and the proof for the static inclusion alone. As an example of this property and the application of the dynamic Eshelby tensor (constant in the interior domain), it is shown that the Eshelby equivalent inclusion method always allows for the determination of the equivalent transformation strain for a self-similarly dynamically expanding inhomogeneous spherical inclusion when the Poisson's ratio is in the real range (positive definiteness of the strain energy). Thus, the solution of dynamically self-similarly expanding inhomogeneities (chemical phase change) with transformation strain can be obtained, as well as the driving force per unit area of the expanding inhomogeneity.

The Eshelby equivalent inclusion method is generalized to calculate the stress fields related to spherical inhomogeneities with two interface conditions depicted by the interface stress model and the linear-spring model. It is found that the method gives the exact results for the hydrostatic loading and very accurate results for a deviatoric loading. The method can be used to predict the effective properties of composites with the interface effects.

Effective moduli of fiber-reinforced polymer matrix composites filled with nanoparticle considering the effect of linear change of interphase are presented in this paper. The three-phase inclusion problem for matrix-interface-particle is equivalent to the Eshelby two-phase inclusion problem. According to the result of the Eshelby inclusion problem, the effective modulus tensor of unit cell of equivalent particle is derived. The effective moduli of equivalent matrix are given based on Mori-Tanaka method. Using two fundamental equation of micromechanic theory, the three-dimensional bridged formulation of unidirectional composites is derived. The quantitative relationship between the macroscopic elastic parameters and the structural parameters of the fiber-reinforced polymer composites filled with nanoparticles is investigated. Effects of the thickness of interfacial layer, the particle size and the volume fraction of nanoparticles on the effective elastic moduli of the composites are also discussed.

This work develops a semi-analytic solution for multiple inhomogeneous inclusions of arbitrary shape and cracks in an isotropic infinite space. The solution is capable of fully taking into account the interactions among any number of inhomogeneous inclusions and cracks which no reported analytic or semi-analytic solution can handle. In the solution development, a novel method combining the equivalent inclusion method (EIM) and the distributed dislocation technique (DDT) is proposed. Each inhomogeneous inclusion is modeled as a homogenous inclusion with initial eigenstrain plus unknown equivalent eigenstrain using the EIM, and each crack of mixed modes I and II is modeled as a distribution of edge climb and glide dislocations with unknown densities. All the unknown equivalent eigenstrains and dislocation densities are solved simultaneously by means of iteration using the conjugate gradient method (CGM). The fast Fourier transform algorithm is also employed to greatly improve computational efficiency. The solution is verified by the finite element method (FEM) and its capability and generality are demonstrated through the study of a few sample cases. This work has potential applications in reliability analysis of heterogeneous materials.

The problem of a semi-infinite crack in anisotropic medium interacting with a near-tip inclusion is analyzed by the Eshelby equivalent inclusion method. The change of mode I stress intensity factor due to crack-inclusion interaction is evaluated using a novel analytical solution for the model I stress intensity factor at the tip of a semi-infinite crack due to near-tip eigenstrains. Numerical results of the mode I stress intensity factor due to the presence of a near-tip circular inclusion are presented to show the influence of the elastic stiffness of an inclusion on the near-tip elastic field. The present scheme can be applied to calculate the stress intensity at a crack-tip in anisotropic media due to the interaction of inclusions with arbitrary shapes.

L’objectif de cette thèse, menée en cotutelle entre l’ISAE-ENSMA / Institut Pprime et l’École de Technologie Supérieure (ETS) de Montréal / LOPFA, était de développer un outil capable de générer un agrégat polycristallin et d’y calculer les champs élastiques locaux en prenant en compte les effets du voisinage de chaque grain. Cet outil a été pensé pour une utilisation statistique afin d’identifier les configurations de voisinage les plus néfastes ainsi que l’influence de la morphologie, de l’anisotropie élastique du matériau et du type de chargement sur ces configurations. S’il existe des modèles capables de calculer les champs mécaniques, ces derniers tendent à minorer les effets de voisinage, et lorsque ces effets sont pris en compte, le coût de calcul est souvent prohibitif. L’outil proposé prolonge les travaux antérieurs de [Bretin et al., 2019] fondés sur l’utilisation des Automates Cellulaires (AC). Tout d’abord, les simulations en champs complets, auparavant nécessaires à l’AC pour calculer l’effet individuel de chaque voisin sur un grain, ont été remplacées par des calculs analytiques via la Méthode de l’Inclusion Equivalente (EIM) [Eshelby, 1957 ; Eshelby, 1959 ; Eshelby, 1961]. Un code EIM unifié permettant de traiter une inhomogénéité élastique, isotrope ou anisotrope (via l’assignation d’une orientation cristallographique (OC)), spatialement orientée dans un milieu isotrope sous chargement uniforme à l’infini, a été spécifiquement développé. Ce code a été validé à chaque nouvelle fonctionnalité introduite en confrontant les champs à l’intérieur et à l’extérieur de l’inhomogénéité selon différents chemins depuis l’interface, à des solutions de référence Éléments Finis (EF). Cette validation systématique pour les points intérieurs et extérieurs garantit la fiabilité du programme final et constitue en soi une contribution originale de la thèse. Le code EIM a ensuite été introduit dans l’AC. Dans un second temps, le modèle de Bretin et l’AC sous-jacent, initialement conçus pour des agrégats réguliers (type Kelvin), ont été enrichis pour pouvoir traiter des agrégats morphologiquement plus réalistes. Un module de génération (Voronoï, Laguerre, Johnson-Mehl) a été introduit à cet effet, ainsi qu’un module pour identifier le voisinage irrégulier de chaque grain et enfin, un module d’approximation des grains par des sphéroïdes inertiellement équivalents. Les résultats du modèle étendu ont été confrontés à des calculs en champs complets par EF pour apprécier leur précision et le gain en temps de calcul est de plusieurs ordres de grandeur inférieur. Dans une dernière partie, le code EIM a été exploité pour étudier l’influence des caractéristiques (forme, rapport de forme, nature, orientations spatiale et cristallographique) sur les contraintes internes et externes à une inhomogénéité isolée dans un milieu infini isotrope. Une deuxième étude , statistique cette fois, a été menée avec des agrégats de 316L puis de Titane, chargés en déformation uniaxiale et en glissement simple. Le rôle de l’anisotropie élastique et de l’orientation cristallographique des voisins est clairement mis en évidence. Des travaux ont été entamés pour insérer une nouvelle phase minoritaire au sein d’un agrégat existant - ici des carbures "apparaissant" dans une microstructure jusque-là chimiquement homogène. Leur effet est important mais localisé
The aim of this PhD thesis, conducted in co-supervision between ISAE-ENSMA / Pprime Institute and the École de Technologie Supérieure (ETS) of Montréal / LOPFA was to develop a tool for the generation of polycrystalline aggregates and the calculation of elastic fields by taking into account the neighborhood effects of each grain. This tool was designed for a statistic employment in order to identify the most harmful neighborhood configurations as well as the influence of the morphology, the elastic anisotropy of the material and of the type of loading on these configurations. Although some existing models allow to estimate local fields, they tend to underestimate the neighborhood effect and when this effect is taken into account, the computational cost is often prohibitive. The advanced tool is an extension of the previous work by [Bretin et al., 2019] based on Cellular Automata (CA). The full-field simulations previously used to compute each neighbor individual effect on a grain were first replaced by analytical calculations with the Equivalent Inclusion Method (EIM) [Eshelby, 1957 ;Eshelby, 1959 ; Eshelby, 1961]. A unified EIM code able to deal with an elastic inhomogeneity, isotropic or anisotropic (via the assignation of a crystallographic orientation (CO)), spatially oriented in an isotropic medium under uniform loading at infinity, was developed to this aim. Each new functionality introduced in the code was carefully validated by comparisons to Finite Element (FE) reference solutions, both inside and outside the inhomogeneity along different paths from the interface. Such a systematic evaluation, for internal and external points, allows to be confident in the reliability of the final program and constitutes an original contribution of the present work. The EIM code was then introduced in the CA. At second, the Bretin’s model and underlying CA, originally devoted to regular aggregates (Kelvin type) were extended in order to deal with more realistic aggregates. A module for the aggregates generation (Voronoï, Laguerre, Johnson-Mehl) was incorporated to this aim as well as a module for the identification of the irregular neighborhood of each grain and at last, a module to approximate the grains by equivalent inertia spheroïds.The results of the extended model were compared to FE full-field calculations in order to appreciate the accuracy and the calculation time several orders of magnitude lower. In a second part, the EIM code was exploited in order to analyse the influence of various characteristics (shape, aspect ratio, nature, spatial and eventual crystallographic orientations) of an isolated inhomogeneity on the inside and outside stress field. A second study, this time statistical, was then carried out with 316L and Titanium aggregates under uniaxial strain loading and shear-strain loading. The role of the elastic anisotropy and of the crystallographic orientation of neighbors was clearly demonstrated. Finally,additional work was undertaken in order to insert a new minority phase into an existing aggregate, here carbides "appearing" in a microstructure previously chemically homogeneous. The effect of these carbides is significant but localized

The interfacial sliding motion of carbon nanotubes (CNTs) within a polymeric hosting matrix gives rise to energy dissipation. By tuning the interfacial shear strength (ISS) of the CNT-matrix interface, the dissipation can take place within tunable ranges of strain amplitudes. This is the basis for conceiving new multilayered carbon nanotube nanocomposites in which different layers with tunable ISS can lead to a concurrent optimization of strength and dissipation, often seen as two conflicting targets. Such optimization is tackled by a novel meso-mechanical nonlinear inelastic model proven to effectively predict the damping capacity of CNT nanocomposites. The proposed elastoplastic, rate-independent, constitutive theory is based on the mean-field homogenization method which combines the Eshelby equivalent inclusion method, the Mori-Tanaka homogenization, and the concept of inhomogeneous inclusions with inelastic eigen strains introduced to describe the inelastic stick-slip. Since the ISS parameter plays a key role in the nanocomposite strength and dissipation, the current work seeks to improve the strength and damping properties by suitable interfacial CNT-matrix functionalization. Variations in the ISS parameter can be achieved by a functionalization that affects the chemical bonds between CNTs and the hosting matrix. A set of experimental tests - including DMA analysis, calorimetry and spectroscopy — aims to evaluate the influence of the ISS parameter, together with other constitutive parameters, on the nanocomposite strength and damping capacity.

This research explores the influence of distributed non-overlapping inhomogeneities on the contact properties of a material. Considered here is the half-space Hertzian contact of a sphere with an inhomogeneous material. The numerical analysis is conducted utilizing a simplified model based on Eshelby’s Equivalent Inclusion Method (EIM) and the principle of superposition. The solutions take into account interactions between all inhomogeneities. Benchmark comparisons with the results obtained with the finite element method (FEM) demonstrate the accuracy and efficiency of the proposed solution methods. The emphasis is given to a parametric study of the effect of inhomogeneities in a Gaussian distribution on material properties. Both compliant and stiff inhomogeneities are modeled. Material inhomogeneities strongly affect rolling contact fatigue (RCF) of a material, and a modified RCF life model is suggested. Homogenization and extensive numerical simulations result in semi-empirical fatigue-life reduction parameters to characterize the influence of material inhomogeneities.

Abstract The effective properties as well as local elastic and electric fields in smart electrostrictive composite materials with periodic microstructure are estimated by using the Fourier series technique (Nemat-Nasser et al., 1993) and the equivalent inclusion method (Eshelby, 1957). The developed model involves many microstructural parameters and can provide guidelines for the design of flexible, nonlinear electrostrictive transducers.

This paper proposes a model to predict the creep response of injection-molded long-fiber thermoplastics (LFTs). The model accounts for elastic fibers embedded in a thermoplastic resin that exhibits the nonlinear viscoelastic behavior described by the Schapery’s model. It also accounts for fiber length and orientation distributions in the composite formed by the injection-molding process. Fiber length and orientation distributions were measured and used in the analysis that applies the Eshelby’s equivalent inclusion method, the Mori-Tanaka assumption (termed the Eshelby-Mori-Tanaka approach) and the fiber orientation averaging technique to compute the overall strain increment resulting from an overall constant applied stress during a given time increment. The creep model for LFTs has been implemented in the ABAQUS finite element code via user-subroutines and has been validated against the experimental creep data obtained for long-glass-fiber/polypropylene specimens. The effects of fiber orientation and length distributions on the composite creep response are determined and discussed.